SOLUTION: What is the rule for finding the number of boxes in a pyramid display that has 100 boxes in the base? The boxes are stacked somewhat like this: x xx xxx xx

Question 96371: What is the rule for finding the number of boxes in a pyramid display that has 100 boxes in the base? The boxes are stacked somewhat like this:
x
xx
xxx
xxxx
Found 2 solutions by checkley71, bucky:
Answer by checkley71(8403) (Show Source): You can put this solution on YOUR website!
THE FORMULA FOR THIS PROBLEM IS (N+1)N/2 WHERE N=THE NUMBER OF ITEMS AT THE BOTTOM.
(100+1)100/2=101*50=5,050 ITEMS ARE IN THE STACK WITH 100 @ THE BASE OF THE PYRAMID.
Answer by bucky(2189) (Show Source): You can put this solution on YOUR website!
For a display that is arranged like a regular pyramid, here's an approach:
.
For the first layer of 100 boxes, the boxes are arranged in a 10 by 10 square.
.
The next layer up will be indented 1 row on all 4 sides. That means that the second layer
will be an 8 by 8 square of boxes.
.
The third layer up will also be indented 1 row from the preceding layer, making it a 6 by 6
layer.
.
The fourth layer is indented 1 row on all sides from the third layer, making it a 4 by 4 layer.
.
The fifth layer is indented 1 row on all sides from the fourth layer, making it a 2 by 2 layer.
.
And the 6th layer is a single box that sits on top of the common intersection of the 4 boxes in
the fifth layer.
.
Adding up all the boxes in the reverse order of what we found to be the size of each layer
results in 1 + 4 + 16 + 36 + 64 + 100 = 221 boxes
.
Notice that this is of the form: 1 + 2^2 + 4^2 + 6^2 + 8^2 + 10^2
.
This tells you that the box contents of a regular pyramid is 1 plus the sum of the squares
of consecutive even numbers until you reach the number of layers you desire. In this problem
the pyramid contained 6 layers. If you wanted to add a seventh layer, what size should the
base be? The answer is that it should be a 12 by 12 base ... requiring 144 boxes. And
an eighth layer would be 14^2 which would be 196 boxes.
.
Hope this makes sense to you. It might help if you sketched a 10 by 10 grid and then indented
1 row on each side for the next layer. That will show you why the next layer is 8 by 8. Then
you can extend on that by indenting 1 row on all sides of the 8 by 8 layer. This will make
it apparent that the next layer has to be 6 by 6. And so on. The final top box is just the
finishing cap to the pyramid, so it is the odd loner in the arrangement.
.

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